Dual-Mode Index Modulation Aided OFDM
TIANQI MAO1, (Student Member, IEEE), ZHAOCHENG WANG1, (Senior Member, IEEE), QI WANG1, (Member, IEEE), SHENG CHEN2,3, (Fellow, IEEE),
AND LAJOS HANZO2, (Fellow, IEEE)
1Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China 2Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K.
3King Abdulaziz University, Jeddah 21589, Saudi Arabia
Corresponding author: L. Hanzo ([email protected])
This work was supported in part by the National Key Basic Research Program of China under Grant 2013CB329203, in part by the National Natural Science Foundation of China under Grant 61571267, in part by Shenzhen Peacock Plan under Grant 1108170036003286, and in part by the Shenzhen Visible Light Communication System Key Laboratory under Grant ZDSYS20140512114229398.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
ABSTRACT Index modulation has become a promising technique in the context of orthogonal frequency division multiplexing (OFDM), whereby the specific activation of the frequency domain subcarriers is used for implicitly conveying extra information, hence improving the achievable throughput at a given bit error ratio (BER) performance. In this paper, a dual-mode OFDM technique (DM-OFDM) is proposed, which is combined with index modulation and enhances the attainable throughput of conventional index-modulation-based OFDM. In particular, the subcarriers are divided into several subblocks, and in each subblock, all the subcarriers are partitioned into two groups, modulated by a pair of distinguishable modem-mode constella-tions, respectively. Hence, the information bits are conveyed not only by the classic constellation symbols, but also implicitly by the specific activated subcarrier indices, representing the subcarriers’ constellation mode. At the receiver, a maximum likelihood (ML) detector and a reduced-complexity near optimal log-likelihood ratio-based detector are invoked for demodulation. The minimum distance between the different legitimate realizations of the OFDM subblocks is calculated for characterizing the performance of DM-OFDM. Then, the associated theoretical analysis based on the pairwise error probability is carried out for estimating the BER of DM-OFDM. Furthermore, the simulation results confirm that at a given throughput, DM-OFDM achieves a considerably better BER performance than other OFDM systems using index modulation, while imposing the same or lower computational complexity. The results also demonstrate that the performance of the proposed low-complexity detector is indistinguishable from that of the ML detector, provided that the system’s signal to noise ratio is sufficiently high.
19 20
INDEX TERMS Orthogonal frequency division multiplexing (OFDM), index modulation, index pattern, constellation, maximum likelihood detection, log-likelihood ratio based detection.
I. INTRODUCTION
21
Orthogonal frequency division multiplexing (OFDM) has 22
become a ubiquitous digital communications technique as a 23
benefit of its numerous virtues, including its capability of 24
providing high-rate data transmission by splitting the serial 25
data into many low-rate parallel data streams [1]. It is also 26
capable of offering a low-complexity high-performance solu-27
tion for mitigating the inter-symbol interference (ISI) caused 28
by a dispersive channel [2], [3]. Due to the above-mentioned 29
merits, OFDM has been adopted in many broadband wireless 30
standards, such as 802.11a/g Wi-Fi, 802.16 WiMAX, and 31
Long-Term Evolution (LTE) [1]. 32
The concept of index modulation (IM) is related to the 33
principle of spatial modulation [4]–[6] originally conceived 34
for multiple-input-multiple-output (MIMO) systems, which 35 was then also invoked in the context of OFDM, leading to the 36 index-modulation-based OFDM philosophy. Explicitly, the 37 information is transmitted using both the classic amplitude 38 as well as phase modulation and implicitly also by the indices 39 of the activated subcarriers [7]–[9]. Index-modulation-based 40 OFDM is capable of enhancing the power efficiency of the 41 classic OFDM, since only a fraction of the subcarriers is 42 modulated, but additional information bits are transmitted 43 by mapping them to the subcarrier domain. Hence various 44 attractive IM-aided OFDM systems have been proposed in 45 the literature. In [10], the specific choice of the activated 46 subcarrier indices conveys extra information in an on-off 47 keying (OOK) fashion, whereby the indices of the activated 48
VOLUME 4, 2016
2169-3536 2016 IEEE. Translations and content mining are permitted for academic research only.
subcarriers are determined by the corresponding majority 49
bit-values of the OOK data streams. Whilst it is capable 50
of attaining a high power efficiency, this scheme imposes 51
a potential bit error propagation, hence leading to bursts of 52
errors. To address this problem, an enhanced subcarrier index 53
modulation OFDM (ESIM-OFDM) scheme was proposed 54
in [11], where each bit of the OOK data streams determines 55
the active subcarrier in a corresponding subcarrier pair, thus 56
only half of the subcarriers are modulated. Although this 57
scheme is capable of enhancing the attainable performance, 58
constellations of higher order are required for modulation 59
in order to achieve the same spectral efficiency as conven-60
tional OFDM. 61
Basar et al. [12] combined OFDM with index modula-62
tion (OFDM-IM) by arranging for the subcarriers to be par-63
titioned into several subblocks, where the specific indices 64
of the active subcarriers in each subblock are used for data 65
transmission. This OFDM-IM scheme is capable of enhanc-66
ing the bit error ratio (BER) performance of classical OFDM 67
at the cost of a reduced throughput, since many subcarri-68
ers remain unmodulated in order to implicitly convey infor-69
mation. In [13] Basar proposed an enhanced version of 70
OFDM-IM by combining space-time block codes with coor-71
dinate interleaving and OFDM-IM, which led to an additional 72
diversity gain. OFDM-IM was also combined by Basar [14] 73
with the MIMO concept, which led to the MIMO-OFDM-IM 74
philosophy, exhibiting a considerable performance gain over 75
classical MIMO-OFDM [15]. Besides, in [16] and [17], the 76
OFDM-IM scheme is introduced to the underwater acous-77
tic communications as well as the vehicle-to-vehicle and 78
vehicle-to-infrastructure (V2X) applications, which achieves 79
significant performance gains. Furthermore, the performance 80
trade-offs of OFDM-IM techniques have been theoretically 81
analyzed in [18]–[21]. Specifically, in [18], a tight BER 82
upper-bound of OFDM-IM was formulated, and the optimal 83
number of active subcarriers was considered in [19] and [20]. 84
In [21], the achievable performance of OFDM-IM and the 85
beneficial region of the OFDM-IM scheme over conven-86
tional OFDM are investigated, which provides the guide-87
lines for system designing of the OFDM-IM scheme. 88
Additionally, several feasible improvements on the perfor-89
mance of OFDM-IM have been discussed in [22] and [23]. 90
Yang et al. [22] proposed a spectrum-efficient index modula-91
tion scheme with improved mapping, allowing each OFDM 92
subblock to use different constellations and different num-93
ber of active subcarriers to ehance the spectral efficiency. 94
While Zheng et al. [23] concentrated on the transceiver 95
design of the OFDM with in-phase/quadrature index modula-96
tion (OFDM-I/Q-IM), which employs both the in-phase and 97
the quadrature components of signals for index modulation. 98
However, the critical problem of throughput loss has not been 99
addressed in the open literature. 100
Against this background, in this paper, a dual-mode OFDM 101
relying on index modulation (DM-OFDM) is proposed, 102
where all the subcarriers are modulated, which enhances 103
the spectral efficiency of the existing OFDM-IM techniques. 104
In our DM-OFDM scheme the subcarriers are partitioned 105 into OFDM subblocks, and for each subblock, information 106 bits are transmitted not only by the modulated subcarriers 107 but implicitly also by the indices of the activated subcar- 108 riers. More specifically, the subcarriers are split into two 109 groups, corresponding to two index subsets. Both groups 110 of subcarriers are then modulated by two different constel- 111 lation modes, and the information bits conveyed by index 112 modulation can be determined by one of the two index sub- 113 sets. At the receiver, a maximum likelihood (ML) detector 114 and a reduced-complexity near optimal log-likelihood ratio 115 (LLR) detector are employed to demodulate the signals. The 116 minimum distance between the different realizations of the 117 OFDM subblocks is calculated in order to evaluate the BER 118 performance of DM-OFDM with the aid of the ML detector. 119 Explicitly, the theoretical analysis is based on the pairwise 120 error probability (PEP) of the proposed DM-OFDM. Our 121 simulation results will demonstrate that DM-OFDM achieves 122 a signal to noise ratio (SNR) gain of several dBs over OFDM- 123 IM both in additive white Gaussian noise (AWGN) channels 124 and in frequency-selective Rayleigh fading channel at the 125 spectral efficiency of 4 bits/s/Hz, while imposing the same 126 or lower computational complexity. Our simulation results 127 will also confirm that the performance of the low-complexity 128 LLR based detector is indistinguishable from that of the 129 ML detector, provided that the system’s SNR is sufficiently 130
high. 131
The rest of this paper is organized as follows. Section II 132 describes the system model of DM-OFDM, while Section III 133 presents our theoretical analysis of the proposed DM-OFDM. 134 Section IV calculates the minimum distance between differ- 135 ent OFDM subblocks, performs the PEP analysis, and carries 136 out Monte Carlo simulations for quantifying the performance 137 of the proposed DM-OFDM, using the existing OFDM-IM as 138 a benchmark. Finally, our conclusions are drawn in Section V. 139
II. DM-OFDM SYSTEM MODEL 140
A. DM-OFDM TRANSMITTER AND CHANNEL MODEL 141 The DM-OFDM transmitter is illustrated in Fig. 1. First, 142 mincoming bits are partitioned by a bit splitter into p groups, 143 each consisting of g bits, i.e., p = m/g. Each group of g 144 information bits is fed into an index selector and two different 145 constellation mappers for generating an OFDM subblock 146 of length l = N/p, where N is the size of fast Fourier 147 transform (FFT). In contrast to the existing index- 148 modulation-based OFDM [12], whereby only part of the sub- 149 carriers are actively modulated, in our DM-OFDM scheme all 150 the subcarriers are modulated in each subblock, which leads 151 to an enhanced spectral efficiency. The first g1 bits of the 152 incoming g bits, referred as index bits, are utilized by the 153 index selector to divide the indices of each subblock into two 154 index subsets, denoted as IAand IB. The remaining g2bits 155 are passed to the mappers A and B having the constellation 156 sets of MA and MB associated with the sizes MAand MB, 157 respectively, where we have MA∧MB= ∅. With the aid of 158 the index selector, the subcarriers corresponding to IAand IB 159
FIGURE 1. System model of the DM-OFDM transmitter.
TABLE 1. A Look-Up Table of Index Modulation For g1=2, l = 4, and k = 2.
are modulated by the mappers A and B, respectively. 160
Assuming that k subcarriers of a subblock are modulated 161
with MA, while the other (l − k) subcarriers are modulated
162
with MB, then g1 and g2 can be calculated respectively 163 according to 164 g1= log2 l! (l − k)!k! , (1) 165 g2= klog2(MA) + (l − k) log2(MB), (2) 166
where b c denotes the integer floor operator. 167
Once IAis known, IBis also determined. Hence we only
168
have to define IA for the index selector. Clearly, there are
169
nI = 2g1 distinctive index patterns, which are given by the
170
index set 171
IA∈I(1)A , IA(2), · · · , I(nAI) . (3)
172
The operations of the index selector are exemplified by 173
Table 1, where we have g1 = 2, l = 4, and k = 2. 174
For each subblock, the indices of the subcarriers modulated 175
by the mapper A are determined by the two index bits, 176
which assume the values of [0, 0], [0, 1], [1, 0] and [1, 1], 177
as I(1)A = [1, 2], I(2)A = [2, 3], I(3)A = [3, 4] and 178
FIGURE 2. An example of DM-OFDM constellation design for MAand MBwith MA=MB=4.
I(4)A = [1, 4], while the other subcarriers are modulated by 179 the mapper B. In order to reliably detect the index pattern 180 at the receiver, the constellations generated by the mappers 181 A and B should be readily differentiable, namely, we have 182 to have MA∧MB= ∅. For example, if quadrature phase 183 shift keying (QPSK) is employed by both mappers, a feasible 184 constellation design is illustrated in Fig. 2. Specifically, an 8- 185 level quadrature amplitude modulation (QAM) constellation 186 is employed, whereby the two sets MA = {−1 −j, 1 − j, 187 1 +j, −1 + j} and MB = {1 + √ 3, (1 + √ 3)j, −1 −√3, 188 −(1 + √
QPSK, respectively. The inner QPSK schemes is associated 190
with mapper A, while the outer one with mapper B. In gen-191
eral, given an (MA+ MB)-QAM constellation, arbitrary MA
192
constellation points can be employed by the mapper A, which 193
are denoted by the set 194
MA=SA(j) : 1 ≤ j ≤ MA , (4)
195
and the other MB constellation points are employed by the
196
mapper B which are denoted by 197
MB=SB(j) : 1 ≤ j ≤ MB . (5)
198
After the mapping, an N -point inverse FFT (IFFT) opera-199
tion is performed following the concatenation of the p differ-200
ent OFDM subblocks 201 X(β) =X(β−1)l+1X(β−1)l+2· · · XβlT 202 =X(β) 1 X (β) 2 · · · X (β) l T, 1 ≤ β ≤ p, (6) 203
to generate the time-domain (TD) signals x =x1x2· · · xN T
, 204
where ( )T denotes the transpose operator. Next the cyclic
205
prefix (CP) of length L is added, and the resultant signals 206
are then fed into a parallel-to-serial converter (P/S) and a 207
digital-to-analog converter (D/A) to generate the transmit 208
signals. Finally, the outputs of the transmitter are transmitted 209
through a frequency-selective Rayleigh fading channel whose 210
channel impulse response (CIR) length v is no higher than the 211
CP length of L. The TD CIR coefficient vector is given by 212
h =h1h2· · · hv
T
, where each hi, 1 ≤ i ≤ v, is a circularly
213
symmetric complex Gaussian random variable following the 214
distribution of CN 0,1v [12]. The frequency domain channel 215
transfer function coefficients (FDCTFCs), defined as the 216
N-point FFT of h, can be represented by 217 H =H1H2· · · HNT = 1 √ NFFT h , (7) 218
where the N -dimensional vector h = h1h2· · · hv0 · · · 0T,
219 and√1
NFFT( ) denotes the N -point FFT operator.
220
At the receiver, after the CP removal, the channel-impaired 221
and noise-contaminated received signals are passed to the 222
N-point FFT processor to yield the frequency-domain (FD) 223
received signal vector Y =Y1Y2· · · YN
T
, which satisfies 224
Yn= HnXn+ Wn, 1 ≤ n ≤ N, (8)
225
where Wn is the FD representation of the AWGN with
226
power N0. In particular, for theβth OFDM subblock, where 227
1 ≤β ≤ p, we have 228
Y(β)=diagX(β) H(β)+W(β), (9) 229
where diagX(β) is the diagonal matrix with its diagonal 230
elements given by the elements of X(β), while 231 Y(β) =Y(β−1)l+1Y(β−1)l+2· · · YβlT 232 =Y(β) 1 Y (β) 2 · · · Y (β) l T , (10) 233 H(β) =H(β−1)l+1H(β−1)l+2· · · HβlT 234 =H(β) 1 H (β) 2 · · · H (β) l T, (11) 235 W(β) =W(β−1)l+1W(β−1)l+2· · · Wβl T 236 =W(β) 1 W (β) 2 · · · W (β) l T . (12) 237
Given each index pattern I(i)A ∈IA, there are nS= MAkM
(l−k)
B 238 legitimate transmit signal vectors for X(β), which is defined 239
by the set 240 S I(i)A = n S(1) I(i)A, S (2) I(i)A, · · · , S (nS) I(i)A o. (13) 241 Each S(j) I(i)A ∈ SI(i)A is an l-dimensional vector S(j) I(i)A = 242 S(j) I(i)A(1) S (j) I(i)A(2) · · · S (j) I(i)A(l) T
. Thus, there are a total of nX = 243 nInS = 2g1MAkMB(l−k) legitimate transmit signal vectors for 244
X(β). 245
It is indicated that the spectral efficiency of the proposed 246 DM-OFDM can be calculated as p(g1+g2)
N +L . According to (1) 247 and (2), the spectral efficiency can be further improved by 248 enlarging the subblock size l. Assuming N is fixed, then the 249 total number of transmitted bits conveyed by index modula- 250 tion becomes nIM = bN/lc
j
log2 klk. To maximize nIM for 251 a fixed l, k is set as l/2 according to the innate property of the 252 combinatorial expression. It is indicated that nIMincreases as 253 lbecomes larger, leading to an improved spectral efficiency 254 for DM-OFDM. However, the computational complexity of 255 the brute-force ML detector is increased exponentially with 256 the size of the OFDM subblocks. Therefore, a trade-off has 257 to be struck between the spectral efficiency and the compu- 258 tational complexity in order to optimize the overall perfor- 259 mance of DM-OFDM, which is set aside for our future work. 260
B. ML DETECTOR FOR DM-OFDM 261 At the receiver, a full ML detector may be invoked for detect- 262 ing the information bits by processing the FD received signals 263 on a subblock by subblock manner. For the βth subblock, 264 the optimal index pattern and transmitted symbols can be 265
obtained by minimizing the ML metric 266
n I(i ∗ ) A , S (j∗) I(i∗)A o =arg min
I(i)A∈IA,S(j)
I(i) A ∈S I(i) A l X k=1 Yk(β)−Hk(β)S(j) I(i)A(k) 2 . 267 (14) 268 After obtaining the ML estimate of both the index pattern 269 and of the symbol vector nI(iA∗), S(j∗)
I(i∗)A
o
, the g = g1 + g2 270 information bits can be demodulated by employing the index- 271 pattern look-up table and the two constellation sets, which 272 map the information bits to the corresponding index pattern 273 and to the constellation symbols. It can be seen from (14) that 274 the computational complexity of the ML detector in terms 275 of complex multiplications is on the order of 2g1Mk
AM
(l−k)
B 276
per subblock, denoted as O 2g1Mk
AM
(l−k)
B . Therefore, the 277
ML detector is impractical to implement for large g1, l, k 278 and modulation orders MAand MB, due to its exponentially 279
C. REDUCED-COMPLEXITY LLR DETECTOR FOR
281
DM-OFDM
282
To reduce the detection complexity, we propose an LLR based 283
detector for DM-OFDM. Since each subcarrier is modulated 284
by either the mapper A or the mapper B, the constellation 285
mode of each subcarrier can be obtained by calculating the 286
logarithm of the ratio between the a posteriori probabilities 287
of the subcarrier being modulated by the mapper A and by 288
the mapper B, respectively, which is formulated as 289 γn=ln PMA j=1Pr(Xn= SA(j)|Yn) PMB q=1Pr(Xn= SB(q)|Yn) ! , (15) 290
where 1 ≤ n ≤ N , SA(j) ∈ MAand SB(q) ∈ MB. It can
291
be seen from (15) that the nth subcarrier is more likely to 292
be modulated by the mapper A if γn is positive, and more
293
likely to be modulated by the mapper B ifγnis negative. Since
294 PMA j=1Pr(Xn= SA(j)) = k/l and P MB q=1Pr(Xn= SB(q)) = 295
(l − k)/l, using Bayes rule, (15) can be further expressed as 296 γn=ln k l − k +ln MA X j=1 exp − 1 N0 |Yn− HnSA(j)|2 297 −ln MB X q=1 exp − 1 N0 |Yn− HnSB(q)|2 . (16) 298
To avoid potential computation overflow, the Jaco-299
bian logarithm [24], [25] is used for calculating the 300 terms lnPMA j=1exp −1 N0 |Yn− HnSA(j)| 2and lnPMB q=1 301 exp−1 N0 |Yn− HnSB(q)|
2. Consider the first term as 302
an example, while the other term can be calculated in 303
the same way. We define λj = −N1
0|Yn− HnSA(j)| for
304
j =1, 2, · · · , MA. The recursion is initialized by
305 ln eλ1+ eλ2 = max{λ 1, λ2} +ln 1 + e−|λ2−λ1| 306 =max{λ1, λ2} + f(|λ1−λ2|) , (17) 307
where f ( ) is known as the correction function. Then given 308 1 = ln eλ1+ eλ2+ · · · + eλχ−1 forχ = 2, 3, · · · , MA, we 309 have 310 ln eλ1+ eλ2+ · · · + eλχ 311 =ln e1+ eλχ 312 =max{1, λχ} +ln1 + e−|λχ−1| 313 =max{1, λχ} + f(|1 − λχ|). (18) 314
By applying (18) iteratively from χ = 2 to MA,
315 lnPMA j=1exp −1 N0 |Yn− HnSA(j)| 2is calculated. Hence, 316
the LLR value ofγncan be obtained using Algorithm 1.
317
After obtaining the signs ofγn, i.e., ¯γn=sgn(γn), for 1 ≤
318
n ≤ N, we can group them into the p blocks as 319 ¯ γ(β) = ¯γ (β−1)l+1γ¯(β−1)l+2· · · ¯γβlT 320 = ¯γ(β) 1 γ¯ (β) 2 · · · ¯γ (β) l T, 1 ≤ β ≤ p. (19) 321
Algorithm 1 Iterative LLR Calculation for DM-OFDM Require: Received signals Yn, FDCTFCs Hn, noise energy
N0, constellation sets MA and MB, and their sizes MA
and MB, size of OFDM subblock l, number of subcarriers
modulated by mapper A per subblock k;
Ensure: γnis LLR of nth subcarrier; 1: 11= −N1 0 |Yn− HnSA(1)| 2; 2: 12= −N1 0 |Yn− HnSB(1)| 2; 3: for (j = 2; j ≤ MA; j + +) do 4: T1= −N10 |Yn− HnSA(j)|2; 5: T2=max{11, T1} + f(|11− T1|); 6: 11= T2; 7: end for 8: for (q = 2; q ≤ MB; q + +) do 9: T1= −N10 |Yn− HnSB(q)|2; 10: T2=max{12, T1} + f(|12− T1|); 11: 12= T2; 12: end for 13: γn=ln(k) − ln(l − k) +11−12; 14: returnγn;
Since ¯γ(β)indicates which subcarriers of theβth subblock are 322 modulated by the mapper A and which subcarriers are mod- 323 ulated by the mapper B, it is equivalent to the index pattern. 324 Thus, ¯γ(β)can be utilized to demodulate the corresponding 325 g1 index bits according to the index pattern look-up table. 326 Furthermore, since ¯γi(β)for 1 ≤ i ≤ l indicates whether Xi(β) 327 is modulated by the mapper A or by the mapper B, the ML 328
estimate for Xi(β)is given by 329
b Xi(β)= arg min SA(j)∈MA Yi(β)− Hi(β)SA(j) 2, ¯ γ(β) l = +1, arg min SB(j)∈MB Yi(β)− Hi(β)SB(j) 2 , γ¯(β) l = −1, 330 (20) 331 where 1 ≤ i ≤ l. This yields the g2estimated information 332
bits for subblockβ. 333
Remark: Under an extremely high noisy condition, it is 334 possible that a ¯γ(β)obtained may not correspond to a legiti- 335 mate index pattern. In such a situation, a solution is to reverse 336 the sign of the LLR having the smallest magnitude in the 337 subblock to see if a legitimate index pattern can be inferred. 338 If the resultant modified ¯γ(β)is still illegitimate, the LLR with 339 the second smallest magnitude can be examined, and so on 340 until a legitimate index pattern is produced [26]. 341 This LLR based detector is clearly a near-ML solu- 342 tion. However, its computational complexity is much lower 343 than that of the ML detector. Specifically, the complexity 344 of calculating the LLRs for a subblock is on the order 345 of O l(MA + MB) in terms of complex multiplications, 346 while the complexity of performing the ML demodula- 347 tion for a subblock given index pattern is on the order of 348 O kMA+(l − k)MB. Therefore, the complexity of this LLR 349 based detector is on the order of O l(MA + MB) per sub- 350 block in terms of complex multiplications. Moreover, in an 351
environment having a sufficiently high SNR, the performance 352
of this LLR detector is indistinguishable from that of the ML 353
detector. Thus the LLR based detector is a better choice for 354
practical DM-OFDM systems. 355
Recently, a novel reduced-complexity receiver was pro-356
posed for the detection of index-modulated OFDM in [27]. 357
Since there is still slight performance gap between the pro-358
posed LLR detector and the ML detector at low SNRs, the 359
reduced-complexity detector in [27] will be also applied in 360
DM-OFDM in our future study, which may harvest on perfor-361
mance gain over the proposed low-complexity LLR detector. 362
III. PERFORMANCE ANALYSIS OF DM-OFDM
363
According to (14), the performance of DM-OFDM relying 364
on the ML detector is determined by the minimum distance 365
between the different realizations of the OFDM subblock. 366
First, let us define the set of nX realizations for the OFDM
367 subblock by 368 Xsb= X(i,j)=S(j) I(i)A : ∀I(i)A ∈IA, S(j) I(i)A ∈S I(i)A , (21) 369
and let us denote the l-dimensional realization vector by 370
X(i,j) = X(i,j)(1) X(i,j)(2) · · · X(i,j)(l)T. The distance metric 371
between two different realization vectors X(i1,j1)and X(i2,j2)
372 is given by 373 D(X(i1,j1), X(i2,j2)) = v u u t l X t=1 X(i1,j1)(t) − X(i2,j2)(t) 2 . (22) 374
Therefore, the minimum distance normalized by the transmit-375
ted bit energy, denoted as dmin, can be formulated as 376
dmin= min
X(i1,j1),X(i2,j2)∈Xsb ∀(i1,j1)6=(i2,j2)
377 × s 1 Eb D X(i1,j1), X(i2,j2) , (23) 378 where Eb = Es(N + L)/m = (Es(N + L)) / (p(g1+ g2)) is 379
the average transmitted energy per bit of DM-OFDM, and Es
380
denotes the average transmitted symbol energy. 381
The performance metric of dmin can also be applied to 382
OFDM-IM if the same ML detector is utilized. Therefore, we 383
can employ the minimum distance of the OFDM subblock as 384
a metric to evaluate the performance of both the DM-OFDM 385
and OFDM-IM. 386
The performance of DM-OFDM using the ML detector 387
can be also estimated by PEP analysis. Since the PEP events 388
in different subblocks are identical [12], analyzing a single 389
OFDM subblock is sufficient. The received signal for the 390
genericβth subblock is given in (9). For notational conve-391
nience, we denote Xdiag=diag{X(β)}. The conditioned
pair-392
wise error probability (CPEP) of the event that the transmitted 393
Xdiagis detected as bXdiagis given by [12]
394 Pr Xdiag→bXdiag|H(β) = Q s Esη 2N0 ! , (24) 395
where Q( ) is the Gaussian Q-function, and 396
η = Xdiag−bXdiagH (β) 2 F = H(β)H 0H(β) (25) 397
in which ( )H denotes the conjugate transpose operator and 398 0 = Xdiag −bXdiagH Xdiag −bXdiag). According to [28], 399
(24) can be approximated as 400 Pr Xdiag→bXdiag|H(β) 401 ≈ 1 12exp − Esη 4N0 +1 4exp − Esη 3N0. (26) 402 Then the corresponding unconditioned pairwise error proba- 403
bility (UPEP) is formulated as 404
Pr Xdiag→bXdiag) 405 ≈ EH(β) 1 12exp − Esη 4N0 +1 4exp −Esη 3N0 , (27) 406 where EH(β){ }denotes the expectation operation with respect 407 to H(β). The correlation matrix of H(β) is defined by C = 408 EnH(β) H(β)Ho. According to [29], C can be decomposed 409 as Q3QH, whereby H(β)=Qv with3 = EnvvHo. Then the
410 probability density function (PDF) of the stochastic vector v 411
can be derived as 412 f(v) = π −r det(3)exp − v H3−1v, (28) 413 where r is the rank of C, and det( ) denotes the determinant 414 operator. Applying (28) to complete the expectation operation 415 in (27) yields the following expression of Pr Xdiag→bXdiag 416
[12] 417 Pr Xdiag→bXdiag 418 ≈ 1 12 detIl+4NEs 0C0 + 1 4 detIl+3NEs 0C0 , (29) 419
where Ildenotes the l ×l identity matrix. After the calculation 420 of the UPEP, the average bit error probability (ABEP) can be 421
derived as [12] 422 Pave= 1 gnX X Xdiag6=bXdiag 423
×Pr Xdiag→bXdiagε Xdiag,bXdiag, (30) 424 whereε Xdiag,bXdiag is the number of bit errors in the cor- 425 responding pairwise error event. Paveis approximately equal 426 to the system’s BER, and therefore it can be used to estimate 427
the performance of DM-OFDM. 428
Although the PEP analysis presented here is based on the 429 ML detection, the result of the approximate BER given above 430 can also be applied to the DM-OFDM relying on the LLR 431 based detector, especially under high-SNR conditions. This 432 is because the reduced-complexity LLR based detector offers 433 a near-ML solution, and its performance is indistinguishable 434
from that of the ML detector at high SNRs. 435
Recently, several improvements based on the conventional 436 OFDM-IM have been proposed [7], [13], [22], [23], which 437
can be also applied to our proposed DM-OFDM. For exam-438
ple, in [7] and [22] OFDM-IM is generalized by allowing 439
each OFDM subblock to have different number of activated 440
subcarriers and different constellation mappings. And both 441
the quadrature and the in-phase components of signals are 442
employed for index modulation in [7] and [23]. Additionally, 443
coordinate interleaving techniques are used in OFDM-IM 444
by Basar [13]. Since both the proposed DM-OFDM and 445
conventional OFDM-IM divide the subcarriers into OFDM 446
subblocks, and perform index modulation within each sub-447
block, the aforementioned enhancements of OFDM-IM can 448
be readily invoked by DM-OFDM. 449
IV. NUMERICAL RESULTS
450
The performance of the proposed DM-OFDM and the 451
existing index-modulation-based OFDM scheme, namely, 452
OFDM-IM, are compared. The frequency-selective Rayleigh 453
fading channel used in the simulation has a CIR length of 454
v =10. The number of subcarriers is set to N = 128, divided 455
into p = 32 subblocks with 4 subcarriers per subblock, and 456
the length of CP is 16. The system’s SNR is defined as Eb/N0. 457
Example 1:The constellations of DM-OFDM are depicted 458
in Fig. 2, whereby 2 subcarriers are modulated by the inner 459
QPSK, and the other 2 subcarriers are modulated by the 460
outer QPSK in each OFDM subblock. Hence, a total of 461
g1+ g2 = 2 + 8 = 10 information bits are transmitted 462
per OFDM subblock, and the spectral efficiency of this DM-463
OFDM is equal to 2.22 bits/s/Hz. For the OFDM-IM coun-464
terpart, 2 subcarriers of each OFDM subblock are modulated 465
by 16-QAM, and the other subcarriers are unmodulated in 466
order to maintain the same spectral efficiency and the same 467
complexity as the DM-OFDM. The dmin metrics for the 468
DM-OFDM and OFDM-IM, denoted as dmin,DM−OFDMand 469
dmin,OFDM−IM, are 1.3706 and 1.3333, respectively. Since we 470
have dmin,DM−OFDM> dmin,OFDM−IM, DM-OFDM achieves 471
a better BER performance than its OFDM-IM counterpart for 472
the same spectral efficiency of 2.22 bits/s/Hz. 473
To validate the above minimum distance based analytical 474
result, the performance comparison of the proposed DM-475
OFDM and the existing OFDM-IM, both using the ML detec-476
tor, is carried out by Monte Carlo simulation. The results 477
obtained are shown in Fig. 3. It can be seen from Fig. 3 that 478
at the BER level of 10−3, the proposed DM-OFDM attains 479
1 dB SNR-gain over OFDM-IM, both for the AWGN and for 480
the frequency-selective Rayleigh fading channels considered. 481
This is because 16-QAM has to be employed in OFDM-482
IM for reaching the same spectral efficiency as DM-OFDM, 483
which is more sensitive both to noise and to interference. 484
This confirms the above theoretical analysis. In this case, 485
both the DM-OFDM using the ML detector and the OFDM-486
IM relying on the ML detector have the same computational 487
complexity, which is on the order of O(1024) per subblock in 488
terms of the number of complex multiplications. Moreover, 489
Fig. 3 characterizes the BER performance of DM-OFDM 490
using the low-complexity LLR based detector. It can be seen 491
that DM-OFDM using the LLR detector only suffers from a 492
FIGURE 3. Performance comparison between DM-OFDM, OFDM-IM and ESIM-OFDM under both AWGN and frequency-selective Rayleigh fading channel conditions for Example 1 with the spectral efficiency of 2.22 bits/s/Hz.
slight performance loss compared to the DM-OFDM relying 493 on the ML detector at low SNRs. When the SNR is suf- 494 ficiently high, the performance of the LLR based detector 495 becomes indistinguishable from that of the ML detector, 496 despite the fact that it imposes a much lower computational 497 complexity than the ML detector, specifically O(32) in com- 498 parison to O(1024) in this example. Furthermore, the BER 499 performance of DM-OFDM is compared to that of the exist- 500 ing ESIM-OFDM arrangement at the spectral efficiency of 501 2.22 bits/s/Hz under the AWGN and the frequency-selective 502 Rayleigh fading channel conditions. It can be readily seen that 503 the proposed DM-OFDM regime achieves 1dB and 3dB per- 504 formance gain over ESIM-OFDM for the AWGN channel and 505 the frequency-selective Rayleigh fading channel respectively, 506 because ESIM-OFDM only activates half of its subcarriers 507 for modulation, hence requiring higher order constellations to 508 reach the spectral efficiency of DM-OFDM, therefore leading 509
to a BER performance loss. 510
Additionally, the theoretical PEP analysis is compared with 511 the simulated BER performance in Fig. 4 for the DM-OFDM 512 under the frequency-selective Rayleigh fading channel 513 considered. At low SNRs, the theoretical analysis becomes 514 inaccurate, because there are several approximations in the 515 PEP calculation, which become inaccurate when the noise 516 is dominant. However, the simulation results agree with the 517 theoretical PEP analysis well when the Eb/N0is above 30 dB, 518 indicating that the PEP analysis of DM-OFDM is more accu- 519
rate at high SNRs. 520
Besides, Fig. 5 presents the complementary cumula- 521 tive distribution functions (CCDFs) for the peak-to-average 522 power ratio (PAPR) of the proposed DM-OFDM and the 523 conventional OFDM-IM with values of N , where the sub- 524 block length equals 4. For DM-OFDM, two subcarriers are 525 modulated by the outer QPSK of Fig. 2, and the other 526 subcarriers are modulated by the inner QPSK. And for 527
FIGURE 4. Comparison between the theoretical PEP analysis and the simulated BER performances for DM-OFDM under the frequency-selective Rayleigh fading channel of Example 1.
FIGURE 5. the CCDF for PAPR of DM-OFDM and OFDM-IM with N equals 128, 256, and 512.
the OFDM-IM counterpart, two subcarriers of each OFDM 528
subblock are modulated by 16-QAM in order to reach the 529
same spectral efficiency as the proposed DM-OFDM. It can 530
be observed that, when the CCDF equals 10−3, the PAPR 531
of DM-OFDM is almost the same as that of OFDM-IM 532
with N = 128, 256, 512. Therefore, although the proposed 533
DM-OFDM may suffer from a slightly higher PAPR than 534
OFDM-IM, the difference can be negligible. 535
Example 2:The constellations of DM-OFDM are depicted 536
in Fig. 6, whereby 2 subcarriers are modulated by the inner 537
16-QAM, while the other 2 subcarriers are modulated by the 538
outer 16-QAM in each OFDM subblock. Therefore, a total 539
of g1+ g2=18 information bits are transmitted per OFDM 540
subblock, yielding the spectral efficiency of 4 bits/s/Hz. For 541
its OFDM-IM counterpart, to reach the same spectral effi-542
ciency as DM-OFDM, 256-QAM is employed to modulate 543
FIGURE 6. DM-OFDM constellation design for MAand MBwith MA=MB=16.
two subcarriers in each OFDM subblock, while the others 544 are unmodulated. The dmin metrics for the DM-OFDM 545
and OFDM-IM are dmin,DM−OFDM = 0.8944 and 546
dmin,OFDM−IM=0.4339. Since the ratio of dmin,DM−OFDMto 547 dmin,OFDM−IMis considerably larger than that of Example 1, 548 the performance gain of the DM-OFDM over the OFDM-IM 549 is expected to be significantly higher for Example 2. 550 Since the computational complexity of both DM-OFDM 551 associated with the ML detector and of OFDM-IM using the 552 ML detector is on the order of O(262144) per subblock in 553 terms of complex multiplications, the ML detection is diffi- 554 cult to implement in practice. However, in order to demon- 555 strate that the performance of the reduced-complexity LLR 556 detector is indistinguishable from that of the ML based detec- 557 tor for sufficiently high SNRs, we implemented both the ML 558 based detector and the LLR based detector for DM-OFDM 559 in our Monte Carlo simulations. Note that the DM-OFDM 560 using the LLR based detector has a complexity on the order of 561 O(128) per subblock, while OFDM-IM using the LLR based 562 detector has a complexity on the order of O(512), which is 563 considerably higher than that of DM-OFDM employing the 564 LLR based detector. Fig. 7 portrays our BER performance 565 comparison of DM-OFDM and OFDM-IM, where it can be 566 seen that at the BER level of 10−3, the DM-OFDM scheme 567 relying on our LLR based detector achieves SNR gains of 568 6 dB and 5 dB over OFDM-IM using the LLR based detec- 569 tor for the AWGN and frequency-selective Rayleigh fading 570 channels respectively, since two 16-QAM constellation sets 571 are invoked by the DM-OFDM, which is naturally more 572 robust both to noise and to interference than OFDM-IM 573 in conjunction with 256-QAM. These large gains are par- 574 ticularly remarkable, considering the fact that in this high 575 spectral efficiency case, the DM-OFDM combined with the 576 LLR based detector actually has a lower complexity than 577 the OFDM-IM with the LLR based detector. The results 578 of Fig. 7 also confirm that the performance loss of the 579
FIGURE 7. Performance comparison between DM-OFDM and OFDM-IM under both AWGN and frequency-selective Rayleigh fading channel conditions for Example 2 with the spectral efficiency of 4 bits/s/Hz.
DM-OFDM using the LLR detector in comparison to the 580
DM-OFDM employing the ML detector is negligible even at 581
low SNRs. Therefore, the reduced-complexity LLR detector 582
is extremely attractive for DM-OFDM systems designed for 583
a high spectral efficiency for its near ML performance and 584
dramatically reduced complexity. 585
Example 3:To demonstrate the superiority of the proposed 586
DM-OFDM over conventional OFDM, two BPSK constel-587
lation sets are adopted in DM-OFDM, which are defined as 588
MA= {1, −1} and MB= {j, −j}. In each OFDM subblock,
589
two subcarriers are modulated by the mapper A, whilst the 590
other subcarriers are modulated by the mapper B. Hence the 591
number of total transmitted bits per subblock is 6, yielding 592
the spectral efficiency of 1.33 bits/s/Hz. For the conventional 593
OFDM counterpart, BPSK are employed for modulation, and 594
the spectral efficiency is equal to 0.89 bits/s/Hz. 595
The performance of the proposed DM-OFDM is com-596
pared with the conventional OFDM counterpart in Fig. 8, 597
where both the frequency-selective Rayleigh fading chan-598
nel and the AWGN channel are considered. It can be seen 599
that the performances of DM-OFDM and the conventional 600
OFDM are almost the same at the BER level of 10−3under 601
AWGN channel assumptions, and the proposed DM-OFDM 602
achieves more than 2dB performance gain over its OFDM 603
counterpart under the frequency-selective channel besides its 604
0.44 bit/s/Hz spectral efficiency gain over OFDM. This is 605
because higher spectral efficiency can reduce Eb, leading to 606
smaller Eb/N0to attain certain BER. Therefore, it is demon-607
strated that the proposed DM-OFDM is capable of enhancing 608
the performance of OFDM and conventional OFDM-IM. 609
Fig. 9 compares the PEP analysis of DM-OFDM using ML 610
detection of Example 3 to the simulated BER performance 611
of DM-OFDM for a frequency-selective Rayleigh fading 612
channel at the spectral efficiency of 1.33 bits/s/Hz. Like in 613
Example 1, at high SNRs, the PEP analysis becomes accurate, 614
as confirmed by the simulated BER. 615
FIGURE 8. Performance comparison between DM-OFDM of 1.33 bits/s/Hz and OFDM-IM of 0.89 bits/s/Hz for Example 3.
FIGURE 9. Comparison between the theoretical PEP analysis and the simulated BER performance for DM-OFDM under the frequency-selective Rayleigh fading channel of Example 3.
Additionally, the theoretical performance based on the PEP 616 analysis of the DM-OFDM under AWGN channel conditions 617 is also compared with its simulated counterpart for Example 1 618 and Example 3, which is illustrated in Fig. 10. It can be seen 619 that, despite the slight performance gap at low SNRs, the 620 theoretical BER results are almost the same as the simulated 621 counterparts at the SNR of 6dB and 10dB at the spectral 622 efficiency of 1.33 bits/s/Hz and 2.22 bits/s/Hz respectively, 623 which are achievable for practical DM-OFDM systems. It is 624 indicated that the PEP analysis under the AWGN channel is 625 more accurate than that of the frequency-selective Rayleigh 626 fading channel condition. This is mainly because the FD 627 channel coefficients are all one under the AWGN channel, 628 and the UPEP can be directly calculated by (26), without 629 the need of taking the approximation of (29), hence reducing 630 the deviations compared with the PEP analysis under the 631
FIGURE 10. Comparison between the theoretical PEP analysis and the simulated BER performance for DM-OFDM under the AWGN channel of Example 1 and Example 3.
V. CONCLUSIONS
633
In this paper, a novel DM-OFDM design has been proposed, 634
whereby the subcarriers are divided into OFDM subblocks 635
and the index modulation technique is applied. However, in 636
contrast to the existing index-modulation-based OFDM, all 637
the subcarriers are modulated, which improves the spectral 638
efficiency. Specifically, at the transmitter, the subcarriers of 639
each subblock are split into two groups, which are modu-640
lated by a pair of distinguishable modulation constellations, 641
respectively. Thus, information bits are conveyed not only by 642
the constellation symbols, but also by the subcarrier indices 643
indicating the constellation modes of the subcarriers. At the 644
receiver, a ML based detector and a reduced-complexity near 645
optimal LLR based detector have been employed to demod-646
ulate the information bits by processing the received signals 647
in the frequency domain. The minimum distance for different 648
realizations of OFDM subblocks has been calculated, which 649
demonstrates that the proposed DM-OFDM outperforms the 650
conventional OFDM-IM, given the same spectral efficiency. 651
Furthermore, the PEP analysis has been performed to esti-652
mate the BER of DM-OFDM. The Monte Carlo simula-653
tion results obtained have validated the theoretical analytical 654
results and have also confirmed that DM-OFDM is capable 655
of achieving a considerably better BER performance than 656
its conventional OFDM-IM counterpart, while imposing the 657
same or lower computational complexity, given the same 658
spectral efficiency. The results have also confirmed that the 659
performance of the DM-OFDM with the LLR based detector 660
is indistinguishable from that of the DM-OFDM with the ML 661
detector when the system’s SNR is sufficiently high. 662
In this paper, we restrict our work on the transceiver design 663
of the proposed DM-OFDM and the corresponding perfor-664
mance analysis. In the future study, we will consider index 665
modulation techniques using tri-mode or quad-mode constel-666
lations, where three or four distinguishable constellation sets 667
are employed for each OFDM subblock, and the throughput 668
can be significantly enhanced. Moreover, we will also inves- 669 tigate the optimization of the size of OFDM subblocks and 670 the constellation design of DM-OFDM, further improving the 671
BER performance. 672
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TIANQI MAO (S’15) received the B.S. degree 760
from Tsinghua University, Beijing, China, in 2015, 761
where he is currently pursuing the M.S. degree 762
with the Department of Electronic Engineering. 763
His current research interests include optical wire-764
less communications and channel coding and 765
modulation. 766
ZHAOCHENG WANG (M’09–SM’11) received 767
the B.S., M.S., and Ph.D. degrees from Tsinghua 768
University, Beijing, China, in 1991, 1993, and 769
1996, respectively. From 1996 to 1997, he was a 770
Post-Doctoral Fellow with Nanyang Technolog-771
ical University, Singapore. From 1997 to 1999, 772
he was with OKI Techno Centre (Singapore) Pte. 773
Ltd., Singapore, where he was first a Research 774
Engineer and later became a Senior Engineer. 775
From 1999 to 2009, he was with Sony Deutschland 776
GmbH, where he was first a Senior Engineer and later became a Princi-777
pal Engineer. He is currently a Professor of Electronic Engineering with 778
Tsinghua University and serves as the Director of Broadband Communi-779
cation Key Laboratory and Tsinghua National Laboratory for Information 780
Science and Technology. He has authored or coauthored over 100 journal 781
papers (SCI indexed). He holds 34 granted U.S./EU patents. He co-authored 782
two books, one of which entitled Millimeter Wave Communication Systems 783
(Wiley-IEEE Press) was selected by the IEEE Series on Digital and Mobile 784
Communication. His research interests include wireless communications, 785
visible light communications, millimeterwave communications, and digital 786
broadcasting. He is a fellow of the Institution of Engineering and Technology. 787
Currently, he serves as an Associate Editor of the IEEE TRANSACTIONSon
788
WIRELESSCOMMUNICATIONSand the IEEE COMMUNICATIONSLETTERS, and has
789
also served as Technical Program Committee Co-Chair of various interna-790
tional conferences. 791
QI WANG (S’15-M’16) received the B.E. (Hons.) 792 and Ph.D. degree in electronic engineering from 793 Tsinghua University, Beijing, China, in 2016 and 794 2011, respectively. From 2014 to 2015, he was a 795 Visiting Scholar with the Electrical Engineering 796 Division, Centre for Photonic Systems, Depart- 797 ment of Engineering, University of Cambridge. He 798 has authored over 20 journal papers. His research 799 interests include channel coding, modulation, and 800 visible light communications. He was a recipient 801 of the Outstanding Ph.D. Graduate of Tsinghua University, Excellent Doc- 802 toral Dissertation of Tsinghua University, National Scholarship, and the Aca- 803 demic Star of Electronic Engineering Department in Tsinghua University. 804 SHENG CHEN (M’90–SM’97–F’08) received the 805 B.Eng. degree in control engineering from the 806 East China Petroleum Institute, Dongying, China, 807 in 1982, the Ph.D. degree in control engineering 808 from City University, London, in 1986, and the 809 D.Sc. degree from the University of Southampton, 810 Southampton, U.K., in 2005. He held research 811 and academic appointments with the University 812 of Sheffield, the University of Edinburgh, and 813 the University of Portsmouth, U.K., from 1986 to 814 1999. Since 1999, he has been with the Electronics and Computer Science 815 Department, University of Southampton, where he is a Professor of Intel- 816 ligent Systems and Signal Processing. He is also a Distinguished Adjunct 817 Professor with King Abdulaziz University, Jeddah, Saudi Arabia. He has 818 authored over 550 research papers. He was an ISI Highly Cited Researcher 819 in engineering in 2004. His research interests include adaptive signal pro- 820 cessing, wireless communications, modeling and identification of nonlinear 821 systems, neural network and machine learning, intelligent control system 822 design, evolutionary computation methods, and optimization. He is a fellow 823 of IET. He was elected to a fellow of the United Kingdom Royal Academy 824
of Engineering in 2014. 825
LAJOS HANZO (F’04) received the D.Sc. degree 826 in electronics in 1976, the Ph.D. degree in 1983, 827 and the Doctor Honoris Causa degree from the 828 Technical University of Budapest in 2009. He also 829 received the Ph.D. degree (Hons.) from the Uni- 830 versity of Edinburgh in 2015. During his 40-year 831 career in telecommunications, he has held vari- 832 ous research and academic positions in Hungary, 833 Germany, and the U.K. From 2008 to 2012, he 834 was the Editor-in-Chief of the IEEE Press and a 835 Chaired Professor with Tsinghua University, Beijing. Since 1986, he has 836 been with the School of Electronics and Computer Science, University of 837 Southampton, U.K., as the Chair in Telecommunications. He has success- 838 fully supervised 110 Ph.D. students, co-authored 20 John Wiley/IEEE Press 839 books in mobile radio communications totaling in excess of 10 000 pages, 840 authored over 1500 research entries at the IEEE Xplore. He has over 25 000 841 citations. He is directing a 100 Strong Academic Research Team, working 842 on a range of research projects in the field of wireless multimedia commu- 843 nications sponsored by the industry, the Engineering and Physical Sciences 844 Research Council, U.K., the European Research Councils Advanced Fellow 845 Grant, and the Royal Societys Wolfson Research Merit Award. He is an 846 enthusiastic supporter of industrial and academic liaison and offers a range of 847 industrial courses. He is also a fellow of the Royal Academy of Engineering, 848 the Institution of Engineering and Technology, and the European Association 849 for Signal Processing. He is a Governor of the IEEE VTS. He acted as the 850 TPC Chair and General Chair of the IEEE conferences, presented keynote 851 lectures, and received a number of distinctions. 852 853
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